Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will learn how to work with locations and directions in three-dimensional space, in order to easily describe objects such as lines, planes and curves. This will set the stage for the study of functions of two variables, the graphs of which are surfaces in space. Points in Three-Dimensional Space Previously, we have identified a point in the xy-plane by an ordered pair that consists of two real numbers, an x-coordinate and y-coordinate, which denote signed distances along the x-axis and y-axis, respectively, from the origin, which is the point (0, 0). These axes, which are collectively referred to as the coordinate axes, divided the plane into four quadrants. We now generalize these concepts to three-dimensional space, or xyz-space. In this space, a point is represented by an ordered triple (x, y, z) that consists of three numbers, an x-coordiante, a y-coordinate, and a z-coordinate. As in the two-dimensional xy-plane, these coordinates indicate the signed distance along the coordinate axes, the x-axis, y-axis and z-axis, respectively, from the origin, denoted by O, which has coordinates (0, 0, 0). There is a one-to-one correspondence between a point in xyz-space and a triple in R 3, which is the set of all ordered triples of real numbers. This correspondence is known as a three-dimensional rectangular coordinate system. Example Figure 1 displays the point (2, 3, 1) in xyz-space, denoted by the letter P, along with its projections onto the coordinate planes (described below). The origin is denoted by the letter O. Planes in Three-Dimensional Space Unlike two-dimensional space, which consists of a single plane, the xy-plane, three-dimensional space contains infinitely many planes, just as two-dimensional space consists of infinitely many lines. Three planes are of particular importance: the xy-plane, which contains the x- and y-axes; the yz-plane, which contains the y- and z-axes; and the xz-plane, which contains the x- and z-axes. Alternatively, the xy-plane can be described as the set of all points (x, y, z) for which z = 0. Similarly, the yz-plane is the set of all points of the form (0, y, z), while the xz-plane is the set of all points of the form (x, 0, z). 1
Figure 1: The point (2, 3, 1) in xyz-space, denoted by the letter P. The origin is denoted by the letter O. The projections of P onto the coordinate planes are indicated by the diamonds. The dashed lines are line segments perpendicular to the coordinate planes that connect P to its projections. Just as the x-axis and y-axis divide the xy-plane into four quadrants, these three planes divide xyz-space into eight octants. Within each octant, all x-coordiantes have the same sign, as do all y-coordinates, and all z-coordinates. In particular, the first octant is the octant in which all three coordinates are positive. Plotting Points in xyz-space Graphing in xyz-space can be difficult because, unlike graphing in the xy-plane, depth perception is required. To simplify plotting of points, one can make use of projections onto the coordinate planes. The projection of a point (x, y, z) onto the xy-plane is obtained by connecting the point to the xy-plane by a line segment that is perpendicular to the plane, and computing the intersection of the line segment with the plane. Later, we will learn more about how to compute projections of points onto planes, but in this 2
relatively simple case, it follows from our working definition that the projection of the point (x, y, z) onto the xy-plane is the point (x, y, 0). Similarly, the projection of this point onto the yz-plane is the point (0, y, z), and the projection of this point onto the xz-plane is the point (x, 0, z). Figure 1 illustrates these projections, and how they can be used to plot a point in xyz-space. One can first plot the point s projections, which is similar to the task of plotting points in the xy-plane, and then use line segments originating from these projections and perpendicular to the coordinate planes to locate the point in xyz-space. The Distance Formula The distance between two points P 1 = (x 1, y 1 ) and P 2 = (x 2, y 2 ) in the xy-plane is given by the distance formula, d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. Similarly, the distance between two points P 1 = (x 1, y 1, z 1 ) and P 2 = (x 2, y 2, z 2 ) in xyz-space is given by the following generalization of the distance formula, d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2. This can be proved by repeated application of the Pythagorean Theorem. Example The distance between P 1 = (2, 3, 1) and P 2 = (8, 5, 0) is d(p 1, P 2 ) = (8 2) 2 + ( 5 3) 2 + (0 1) 2 = 36 + 64 + 1 = 101 10.05. Equations of Surfaces In two dimensions, the solution set of a single equation involving the coordinates x and/or y is a curve. In three dimensions, the solution set of an equation involving x, y and/or z is a surface. Example The equation z = 3 describes a plane that is parallel to the xy-plane, and is 3 units above it; that is, it lies 3 units along the positive z-axis from the xy-plane. On the other hand, the equation x = y describes a plane consisting of all points whose x- and y-coordinates are equal. It is not parallel to any coordinate plane, but it contains the z-axis, which consists of all points whose x- and y-coordinates are both zero, and it intersects the xy-plane at the line y = x. The equation of a sphere with center C = (h, k, l) and radius r is (x h) 2 + (y k) 2 + (z l) 2 = r 2. The unit sphere has center O = (0, 0, 0) and radius 1: x 2 + y 2 + z 2 = 1. 3
We now illustrate how to work with equations of spheres. Example The equation of a sphere with center C = ( 3, 1, 1) and radius r = 10 is or Expanding, we obtain (x ( 3)) 2 + (y ( 1)) 2 + (z 1) 2 = 10 2, (x + 3) 2 + (y + 1) 2 + (z 1) 2 = 100. x 2 + y 2 + z 2 + 6x + 2y 2z = 89, which obscures the center and radius, but it is still possible to detect that the equation represents a sphere, due to the fact that the x 2, y 2 and z 2 terms have equal coefficients. Example The equation 4x 2 + 4y 2 + 4z 2 8x 16y 16 = 0 describes a sphere, as can be seen by the equal coefficients in front of the x 2, y 2 and z 2. determine the radius and center of the sphere, we complete the square in x and y: 0 = 4x 2 + 4y 2 + 4z 2 8x 16y 16 = 4(x 2 2x) + 4(y 2 4y) + 4z 2 16 = 4(x 2 2x + 1 1) + 4(y 2 4y + 4 4) + 4z 2 16 = 4[(x 1) 2 1] + 4[(y 2) 2 4] + 4z 2 16 = 4(x 1) 2 + 4(y 2) 2 + 4z 2 36. Rearranging, we obtain the standard form of the equation of the sphere: (x 1) 2 + (y 2) 2 + z 2 = 9, which reveals that the center is at the point C = (1, 2, 0), and the radius is r = 3. To 4
Summary The three-dimensional rectangular coordinate system is the one-to-one correspondence between each point P in three-dimensional space, or xyz-space, and an ordered triple (x, y, z) in R 3. The numbers x, y and z are the x-, y- and z-coordinates of P. The origin O is the point with coordinates (0, 0, 0). The coordinate planes are: the xy-plane, the set of all points whose z-coordinate is zero; the yz-plane, the set of all points whose x-coordinate is zero; and the xz-plane, the set of all points whose y-coordinate is zero. The projection of a point P = (x, y, z) onto the xy-plane is the point (x, y, 0). The projection of P onto the yz-plane is the point (0, y, z). The projection of P onto the xz-plane is the point (x, 0, z). The distance formula states that the distance between two points in xyz-space is the square root of the sum of the squares of the differences between corresponding coordinates. That is, given P 1 = (x 1, y 1, z 1 ) and P 2 = (x 2, y 2, z 2 ), the distance between P 1 and P 2 is given by d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2. The equation of a sphere with center C = (x 0, y 0, z 0 ) and radius r is (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2. An equation in which x 2, y 2 and z 2 have the same coefficients describes a sphere; the center and radius can be determined by completing the square in x, y and z. 5